Sharp decay rates for a class of nonlinear viscoelastic plate models

2017 ◽  
Vol 20 (02) ◽  
pp. 1750010 ◽  
Author(s):  
E. H. Gomes Tavares ◽  
M. A. Jorge Silva ◽  
T. F. Ma
Keyword(s):  
Decay Rate ◽  
Evolution Equations ◽  
Viscoelastic Plate ◽  
Energy Decay ◽  
Decay Rates ◽  
Memory Kernel ◽  
Energy Decay Rate ◽  
Plate Equations ◽  
Energy Decay Rates ◽  
With Memory

This paper is concerned with uniform stability of the energy corresponding to a class of nonlinear plate equations with memory. It is assumed that the memory kernel [Formula: see text] satisfies the condition [Formula: see text] of Alabau-Boussouira and Cannarsa [A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I 347 (2009) 867–872], where [Formula: see text] is positive, convex, increasing, and satisfies [Formula: see text]. Then, we obtain sharp energy decay rate in the sense that it recovers the decay rate assumed to the memory kernel. To this end we use a recent approach proposed by Lasiecka and Wang [Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer Series INDAM, Vol. 10 (Springer, 2014), pp. 271–303].

scholarly journals Optimal energy decay rates for abstract second order evolution equations with nonautonomous damping

2021 ◽  
Author(s):  
J. R. Luo ◽  
T. J. Xiao
Keyword(s):  
Evolution Equation ◽  
Polynomial Growth ◽  
Evolution Equations ◽  
Energy Decay ◽  
Second Order ◽  
Decay Rates ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Energy Decay Rates ◽  
Autonomous Evolution

We consider an abstract second order non-autonomous evolution equation in a Hilbert space $H:$ $u''+Au+\gamma(t) u'+f(u)=0,$ where $A$ is a self-adjoint and nonnegative operator on $H$, $f$ is a conservative $H$-valued function with polynomial growth (not necessarily to be monotone), and $\gamma(t)u'$ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient $\gamma(t)$ and the exponent associated with the nonlinear term $f$? There seems to be little development on the study of such problems, with regard to {\it non-autonomous} equations, even for strongly positive operator $A$. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of $\gamma(t)$ and $f$.  As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when $f$ is a monotone operator.

ENERGY DECAY OF ELECTROMAGNETIC SYSTEMS WITH MEMORY

2005 ◽  
Vol 15 (10) ◽  
pp. 1489-1502 ◽  
Author(s):  
CLAUDIO GIORGI ◽  
MARIA GRAZIA NASO ◽  
VITTORINO PATA
Keyword(s):  
Initial Data ◽  
Decay Rate ◽  
Energy Decay ◽  
Evolution Problem ◽  
Memory Kernel ◽  
Linear Evolution ◽  
Exponentially Stable ◽  
Systems With Memory ◽  
With Memory ◽  
Linear Evolution Problem

We consider a linear evolution problem with memory arising in the theory of hereditary electromagnetism. Under general assumptions on the memory kernel, all single trajectories are proved to decay to zero, but the decay rate is not uniform in dependence of the initial data, so that the system is not exponentially stable. Nonetheless, if the kernel is rapidly fading and close to the Dirac mass at zero, then the solutions are close to exponentially stable trajectories.

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